On similarity and pseudo-similarity solutions of Falkner-Skan boundary layers

TL;DR

This paper explores similarity and pseudo-similarity solutions of Falkner-Skan boundary layers, identifying conditions for their existence and analyzing solution behaviors, including oscillations and monotonicity, with numerical validation.

Contribution

It introduces new exact pseudo-similarity solutions for specific external velocity profiles and analyzes their properties, including existence conditions and solution behaviors.

Findings

Solutions exist only for a specific suction parameter.

An infinite number of solutions, including oscillatory and monotonic, are established.

Numerical solutions confirm theoretical analysis using Runge-Kutta and shooting methods.

Abstract

The present work deals with the two-dimensional incompressible,laminar, steady-state boundary layer equations. First, we determinea family of velocity distributions outside the boundary layer suchthat these problems may have similarity solutions. Then, we examenin detail new exact solutions, called Pseudo--similarity, where the external velocity varies inversely-linear with the distance along the surface $ (U_e(x) = U_\infty x^{-1}). The present work deals with the two-dimensional incompressible, laminar, steady-state boundary layer equations. First, we determine a family of velocity distributions outside the boundary layer such that these problems may have similarity solutions. Then, we examenin detail new exact solutions. The analysis shows that solutions exist only for a lateral suction. For specified conditions, we establish the existence of an infinite number of solutions, including monotonic solutions and solutions which oscillate an infinite number of times and tend to a certain limit. The properties of solutions depend onthe suction parameter. Furthermore, making use of the fourth--order Runge--Kutta scheme together with the shooting method, numerical solutions are obtained.